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arXiv:quant-ph/0511125 · Symmetry Integrability and Geometry Methods and Applications

Quantum Potential and Symmetries in Extended Phase Space

S. Nasiri

Published Jun 27, 2006• 3 citations
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Abstract

The behavior of the quantum potential is studied for a particle in a linear and a harmonic potential by means of an extended phase space technique.This is done by obtaining an expression for the quantum potential in momentum space representation followed by the generalization of this concept to extended phase space.It is shown that there exists an extended canonical transformation that removes the expression for the quantum potential in the dynamical equation.The situation, mathematically, is similar to disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates that changes the physical potential to an effective one.The representation where the quantum potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form, is one in which the dynamical equation turns out to be the Wigner equation.

DOI
10.3842/sigma.2006.062

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The behavior of the quantum potential is studied for a particle in a linear and a harmonic potential by means of an extended phase space technique.This is done by obtaining an expression for the quantum potential in momentum space representation followed by the generalization of this concept to extended phase space.It is shown that there exists an extended canonical transformation that removes the expression for the quantum potential in the dynamical equation.The situation, mathematically, is similar to disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates that changes the physical potential to an effective one.The representation where the quantum potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form, is one in which the dynamical equation turns out to be the Wigner equation.
In a previous work the concept of quantum potential is generalized into extended phase space (EPS) for a particle in linear and harmonic potentials.It was shown there that in contrast to the Schrödinger quantum mechanics by an appropriate extended canonical transformation one can obtain the Wigner representation of phase space quantum mechanics in which the quantum potential is removed from dynamical equation.In other words, one still has the form invariance of the ordinary Hamilton-Jacobi equation in this representation.The situation, mathematically, is similar to the disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates.Here we show that the Husimi representation is another possible representation where the quantum potential for the harmonic potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form.This happens when the parameter in the Husimi transformation assumes a specific value corresponding to Q-function.
Here, the concept of quantum potential that has been illustrated in Extended Phase Space (EPS) in previous works is explored for its marginal behavior. Unlike in configuration space, different representations of the quantum mechanics can be found in EPS when exploiting appropriate canonical transformations. These canonical transformations revealed that there exist several representations in which the quantum potential could be removed not only for a particle in constant, linear and quadratic potentials, but also for a free particle. In these representations one still has the form-invariance of the ordinary Hamilton-Jacobi equation. The present work is an attempt to show that the Squeezed Wigner representation is a possible representation in which, for large values of squeezing parameter, the quantum potential disappears from the dynamical equation for a particle in the general potential
The composition of the quantum potential and its role in the breakdown of classical symplectic symmetry in quantum mechanics is investigated. General expressions are derived for the quantum potential in both configuration space and momentum space representations. By comparing the configuration space and momentum space representations of the causal interpretation of quantum mechanics, the quantum potential is shown to break the symplectic symmetry that exists between these two representations in classical mechanics. In addition, it is shown that the quantum potential in configuration space may be expressed as the sum of a momentum dispersion energy and a spatial localisation energy; a complementary expression for the quantum potential being found in the momentum representation. The composition and role of the quantum potential in both representations is analysed for a particle in a linear potential and for two eigenstates of the quantum harmonic oscillator.
This paper investigates the composition of the quantum potential and explores its role in the breakdown of symplectic symmetry in quantum mechanics. General expressions are derived for the quantum potential in both configuration space and momentum space representations. By comparing the configuration space and momentum space representations of the causal interpretation of quantum mechanics, the quantum potential is shown to break the symplectic symmetry that exists between these two representations in classical mechanics. It is shown that the quantum potential in configuration space may be expressed as the sum of a momentum dispersion energy and a spatial localisation energy. The quantum potential is thus shown to represent the solely quantum aspects of quantum systems. The composition and role of the quantum potential is analysed for a particle in a linear potential and for two eigenstates of the quantum harmonic oscillator.
Gauge transformations of the potentials for a prescribed electromagnetic field acting on a non-relativistic charged particle are shown to induce canonical transformations in the extended phase space, in which the time and its conjugate momentum are treated as dynamical variables. The extended Hamiltonian is invariant and the extended canonical equations are covariant under a gauge transformation. The usual time-dependent wave equations in different gauges are obtained from the formalism by using Schrödinger representations.